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Taguchi's loss function

Taguchis loss function has no sound mathematical basis, it was derived by simply arbitrarily truncating Taylor’s theorem at a mathematically convenient point. The loss function is often mis-applied by defining the loss function at points where quality losses are known and then interpolating these costs in a region where these costs are not applicable. For example it is irrational to interpolate warranty costs (even a fraction off) inside a region where warranty complaints do not occur. Grandiose claims of cost savings are often made as a consequence of the application of the loss function.

Below I will relate the loss function to specifications.

That quality itself does not follow a go-no go step function as implied by specifications is probably true in most situations. However this does not mean that the concept of specifications should be discredited. We often hear that it is no longer good enough to produce specification conforming products and that variability should be reduced to well inside specifications.

Taguchis loss function is only a partial loss function which considers only the loss the product imparts to society after it is shipped. (Some now include losses in manufacturing due to deviation from the optimum.) It does not include the costs required to maintain product close on target, which are also a loss to society.

Just as it has been accepted by the quality profession that appraisal and prevention costs must be balanced with failure costs, so Taguchi’s losses (analogous to failure costs) must be balanced against losses incurred due to operating at reduced variability (analogous to prevention costs). In general it is reasonable to assume that manufacturing costs increase as variability about the target is reduced. Process control must be tighter, capital may need to be spent, which could have been invested, etc. Notwithstanding the dangers of using general functions, for the sake of argument a loss curve such as shown by curve (1) in Figure 1, could be considered a general conceptual case for losses incurred due to operating away from target. Similarly curve (2) could be considered a general conceptual case to describe the losses incurred by reducing variability about the target. The total loss function, shown by the dotted line and in Figure 2, is obtained by summing both costs.

FIGURE 1: COMPONENT LOSS FUNCTIONS
FIGURE 2: THE TOTAL LOSS FUNCTION

An optimum level of variability is obtained. If variability is allowed to increase beyond this point the losses due to failure and customer dissatisfaction outweigh the losses due to maintaining variability beyond this point. If variability is reduced to within the optimum, the losses due to maintaining variability close to target outweigh the losses caused by the variability about the target. Society loses because resources channelled into tight control could have been allocated to more beneficial causes.

The latter contradicts the Taguchi “school of thought” which encourages tightening variability well within engineering specifications, without regard to costs. The total loss function concept demonstrates the importance of careful specification setting. Not only does it discourage excessively wide specifications but also overly tight specifications. This is often not appreciated in industry, where there is a tendency to set specifications as tight as possible, irrespective to economic considerations.

This must not imply that there is no scope for continuous improvement and reducing variability. If process control costs can be reduced or failure costs increase, then further tightening may be economical. The total loss function merely emphasises the importance of taking a balanced approach to decision making.

CONCLUSION

Although there may be some appropriate applications, there are dangers (more than discussed) in the practical application of the loss function. The loss function is probably best used as a teaching tool to explain to students the consequences of variability. However such teachings should also include the total loss function concept.

Quality professionals are advised to be cautious in the application of ideological and revolutionary concepts. It is best to take each situation as it comes and then decide the best course of action through the normal steps of problem solving, within operational constraints. This need not be traumatic and indeed can be fun.